Question

Write the polynomial in standard form and also write down their degree.
     \[4{{z}^{3}}-3{{z}^{5}}+2{{z}^{4}}+z+1\].

Answer 1

Hint: Focus on the degree of the terms rather than the coefficient. Standard form means arranging the terms in the order of highest power to lowest power. If we arrange it on the basis of the co-efficient then we will end up in a form that is not standard. So, arrange it in a manner such that the term with highest power comes first and the power is in descending order from first to last.

Complete step-by-step answer:
A polynomial is an expression which consists of variables and co-efficient and they have various operations of addition, subtraction, multiplication etc. between them
In the term given in the question we see 5 individual terms connected by addition and subtraction signs.

The first step of rearranging the terms into the standard form is to check for its highest power which is the superscript attached to each term.
The highest power in the term is 5 and the whole term is c.
Now we search for the rest of the powers and rearrange them in the decreasing order

Next, we have the power of 4 and the whole term is$2{{z}^{4}}$.
After that we have power of 3 with terms as$4{{z}^{3}}$.
Now the power of 2 is missing so we can skip it and go to powers of 1 and 0 and the terms are z and 1 respectively.
So, after arranging terms in the standard form we get$-3{{z}^{5}}+2{{z}^{4}}+4{{z}^{3}}+z+1$.

Note: We should avoid getting confused between the co-efficient and degree of a term. No matter how big the co-efficient is or whatever sign it has, our first term will always be of that which has the highest power i.e. Highest degree.